Functional inequalities for the decay of sub-Markov semigroups
type: Item
instance of: scholarly article
title: Functional inequalities for the decay of sub-Markov semigroups
zbMATH Open document ID: 1022.47029
DOI: 10.1023/A:1020535718522
author: Feng-Yu Wang
published in: Potential Analysis
publication date: 2003-04-27T00:00:00Z
review text: Let \((E,{\mathcal F},\mu)\) be a measure space and \(P(t)\) a strongly continuous semigroup on \(L^2(\mu)\) with generator \(L\). The author investigates relationship between the following estimates: \[ \mu (P(t)f)^2) \leq \zeta(t) \Psi(f), \quad t > 0,\;f\in L^2(\mu), \] \[ \mu(f^2) \leq \alpha(r) \mu(fLf) +r\Phi(f), \quad r>0,\;f\in{\mathcal D}(L), \] where \(\zeta:(0,\infty)\to(0,\infty)\) is decreasing with \(\zeta(t)\to 0\) as \(t\to\infty\); \(\alpha:(0,\infty)\to(0,\infty)\) is decreasing; \(\Phi:L^2(\mu)\to[0,\infty]\), \(\Phi(cf)=c^2\Phi(f)\), \(c\in{\mathbb R}\); \(\Psi:L^2(\mu)\to[0,\infty]\), \(\Psi(cf)=c^2\Psi(f)\), \(c\in{\mathbb R}\). The results are illustrated by examples of diffusion and jump processes. Cheeger's inequality for the principal eigenvalue is generalized.
reviewed by: Michael Perelmuter
Mathematics Subject Classification ID: 47D07
zbMATH DE Number: 1901230
zbMATH Keywords: strongly continuous semigroup
zbMATH Keywords: diffusion processes
zbMATH Keywords: Cheeger's inequality
MaRDI profile type: Publication
full work available at URL: a:1020535718522
OpenAlex ID: W159300446
description: scientific article; zbMATH DE number 1901230